# Chapter 13: Chapter 13

Problem 94

Explain why, in computing the total change of a quantity from its rate of change, it is useful to have the definite integral subtract area below the \(x\) -axis.

Problem 94

Give an example of an integral that can be calculated either by using the power rule for antiderivatives or by using the substitution \(u=x^{2}+x\), and then carry out the calculations.

Problem 95

If \(f(x)\) is a continuous function defined for \(x \geq a\), define a new function \(F(x)\) by the formula $$ F(x)=\int_{a}^{x} f(t) d t $$ Use the Fundamental Theorem of Calculus to deduce that \(F^{\prime}(x)=f(x) .\) What, if anything, is interesting about this result?

Problem 95

Show that none of the following substitutions work for $\int e^{-x^{2}} d x: u=-x, u=x^{2}, u=-x^{2} .\( (The antiderivative of \)e^{-x^{2}}$ involves the error function \(\left.\operatorname{erf}(x) .\right)\)

Problem 96

Show that none of the following substitutions work for $\int \sqrt{1-x^{2}} d x: u=1-x^{2}, u=x^{2}\(, and \)u=-x^{2} .$ (The antiderivative of \(\sqrt{1-x^{2}}\) involves inverse trigonometric functions, discussion of which is beyond the scope of this book.)